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Global Existence, Exponential Decay and Blow-Up of Solutions for a Class of Fractional Pseudo-Parabolic Equations with Logarithmic Nonlinearity

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Submitted:

15 January 2018

Posted:

22 January 2018

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Abstract
In this paper, we study the fractional pseudo-parabolic equations ut + (−△)s u + (−△)s ut = u log |u|. Firstly, we recall the relationship between the fractional Laplace operator (−△)s and the fractional Sobolev space Hs and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of global weak solution: for the low initial energy J(u0) < d, the solution is global in time with I(u0) > 0 or ∥u0X0(Ω) = 0 and blows up +∞ with I(u0) < 0; for the critical initial energy J(u0) = d, the solution is global in time with I(u0) ≥ 0 and blows up at +∞ with I(u0) < 0. The decay estimate of the energy functional for the global solution is also given.
Keywords: 
blow-up; fractional pseudo-parabolic equations; initial energy; logarithmic nonlinearity
Subject: 
Computer Science and Mathematics  -   Analysis
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

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